#494846

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,11,15,10,14,25,30,9,13,17,29,24,28,31,35)(7,16,19,8,18,23,32,22,20,12,26,21,33,27,36,34);(5,7,17,19,10,18,31,32,9,20,11,26,24,33,25,36)(6,12,15,8,14,27,30,22,13,16,29,21,28,23,35,34);(2,4)(5,8,9,21)(6,7)(10,22,24,34)(11,27,17,23)(12,25,16,31)(13,20)(14,18)(15,26)(19,29)(28,33)(30,36)(32,35);(6,13)(8,21)(12,16)(14,28)(15,29)(22,34)(23,27)(30,35)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=ch,e^b=ei,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402042);